Philip Hall Lecture Notes on Group Theory – Part V

Philip Hall Lecture Notes on Group Theory – Part V

Hall, Philip, 1904-1982

This material is a set of unpublished, hand-written lecture notes composed by Philip Hall (1904-1982). They were written in the 1960s at the University of Cambridge, where Hall was a professor at King’s College.

Part V

§5 p-groups. Sylow subgroups. (pp. 30-36)

Hall, Philip, 1904-1982

The pages belong to Edward Spitznagel, Washington University Professor of Mathematics, and are xerographic reproductions made from originals held by (and with permission of) John G. Thompson, a contemporary and correspondent of Hall.

1951-1967

Washington University asserts no copyright over digital reproductions of this material, and cannot give permission to use the materials or otherwise grant any exclusive rights of the copyright holder. It is your responsibility to assure that your use of the materials is permitted under applicable law.

K.

Let ; k, ~ ;~ ,.

(A)

~

:.a

~

A . ~ .Jl~ ortiu-_;.. ~ ;~ ~ t

}>.-~ •

p-~s /,.,~ ""-.~1

t;c.c.C...l. ;~c.~

~J.. J... "~ ~
to ~ ~.r Jc~~ . . A sa-o-J ~~ "'J. !tio~ ~ fro-,
L~~ s-1
L.J ~ ;- ~ G- 6c. H..,~ t:~ r-,_~~
1 / ~ sd X Cl"J lJ: ~ ~ It:.... sJ: 1 tJJ. -c.~...._...JA tj X .Jiv.·J.
'lrc.
~ ~~~Mio..,.k l,.:J G.
~ I XI = /Y/ ~ j> •
F Ld' ~,- ·s- ,, if X;_ ;,. ~ of lk., tM"'
,..~"J-.:vJ:t c~ ;_ 1~
~
~ '1 G ~ ft.,.. IX~ I ~ lt.L ~ ttJ G-. : S~ /G-/ ..:.
a.
~~r 1- ; , ~ I X,; I -= J.. ar .Js... I X.:/ :. k'<l'it;)l.,
t.
He...CA, I XI ~ 2: /X,:I a /Y/ .......J; ~ S~a, /Y/ ~ ~ "-'•Lc.r 1~ X;. ~ ~- s.:..sl.. d.... ..,..t .
~s--2.
Ld .G., ~e a- r-.~ ·
.

':t

W I<
: If

<t G-

G i= 1.

fv-e~

~

fij_,.

k

Ev-e.7- s~

ey; )

£~7- ~~

(viiJ

E~"J

cJ...Aj

. ,.~ .tl.~ 1:t· lf ~

~c..c. k

({liJ

1<

i..

f

I<

A

A ... J

~.rr

+

J..,_. a.-rk.r f.
s., 1r ~ KJ'f+t.CJ--:t~

~- ~

:reS-#

NG (H) .

~CKCA- """~

• ·

G-

fK

"f

~:J a.J~t..Y-c. / ( n ¥G . i6
&

J~ .-J; c.-:1-A~

(!_I )

/ '*-CA-

1 G- • 74~ H <
-J ~rk.x ; u . . ~ .;. 6-.
s~ 1- G ·w i .:..~ f _,

v> Arr0

f
~~
cf K .

"f G- A~ ~r f

~~r ..r~

o...

ty)

L;r

I< =F 1. . ~.... l<. '"' JIG 4- 1 .
}'G- -:p 1 .

~~ .... ~ .s~

· L~.t H

(jv)

Aoo.J

}

/"\ -} &--- -:I<

-=/=-

1 . B4 IX/ =/1<1-=/',. ~ .r~-r;;,o
f.l.e.u- I y I ~ ..c ~t J> ~ ~.:J 5 ·I •

1 .·

lt... s~

CA4.c.

K=G

1-

(!) •

K ~ _ A.~~~ J~ ?' G

G- =/=

L.

[j.,.k ~~ .f~

~-v. . ~

r

"if } G
-tJ Jt.., ~~ 1 K.
.

I

I

\

'

iJ,

1

>

fLu.. K * l ·

~~ ~ G-.

/

~ "-

H ~ ~ , . . ~ a-J /& \ H/-==-f,.) ....:JJ. .,..;,0 $~ H< Gr- 1k, '/~.;t,.;._ "it "' 6- ., ~ ~ ; .,.. . Art':!
1. ls ~ HAT~Jc.:f 'f 1ft lo H
7lc~ X .;, ltc, .ret 1- .JL ~/; or H ~ G- 4AcJ
BJ

(j:f)

2.:'1-,

#

s
0

o

a.~

I~~

H ;J-.ul{- '. w-ll.J.

£;..ci I X I = ,~;)

it

~

io

~

s

ir..'\fcaria...t

I yI

/4)

,_d..,r H

~ ~t

t

>

{yrr-

1+:-..CA, H ~ H -= H
{«- Sr-&. ~,_,,..t S 1 G- ~
?L~ k-fL.i.. lt.i: I H l-l ~ H Q...J lo t-f E ' N& (H)
c.n:.
lY)

HH = H .

MftCQ..

;1: ~ A.oC

i. »t~

. s~

,_:..,

Zu-r, -

H.

I ; H)

!C.. ~J H < N<Y(.H).
c:a...J lri J ev-e. ~~ ~ 1- V" J •

(yji)

~

f'cn- (iii)

s~

A..

d¥ f«.d"«"

H/1<.

1

G

.;.

o..

-.:-.;_,;_

'n~ · s~ 1 lk, r-p&uf G-/1<. •
·
.
) ~ A' s.e.d-~ Lf M
4.~ ~ G- i. c .. &J "- c:.ct.d-,J. f-=d-trr ~
G- f M <J G ~ ,LjM ~ }(G-/M) . 7L:- ~l..A.o ~ L<J G~
s-. .2 l!1iJ s~ lt:..k ~ ~ .}!-f!!"t G , ~~ cl.¥ (-=¢err- ~

-r

4-

~,..J_

f.=da-r o

esJkJ ~;g,~ 'f il. A..o 4.. ~ wt~
fo-el-~ ~ o1L ~r-l (-,JIIN 1 G.
I~ oJ.J.; />-~ ~ 1\.·+~-t.
@) A-,~~ ~ ; ~ 1 s, 2 (!·.,.)
we. .,..~ /u,.re,
'
)

A

~ G

~

Jl

L((.~s~ i

lte.""

If ·. H.:. A":/ .r~ tf 4. ~ G- ~~ N-=Ntr(H).~
1/.a, ~ralu.u- "f fH L
lT) .;.. ~ (G-) Cth'wO '-o 14 ?- a1J_ lt..., ~~

.

~f~
~

. -i.H (o<) : H!
o{ EN .
If ia

---'>

0(

HI

HO(

-=.

;,.~~ .,.:JJ:.

fA... -n.u-k.r of- ~""~ 1 H _lA.,
. lJ.e ~ ~ - ~ IW:s-o ~ /N: H/ u ~ "~ of
;H .
If 0( E N .> ltk- C( H =- H o{
~J~
/

s

(Sf G-)

NjH w
;,/;, c:rrJ..r /N: H} ..:c.
~ te;~..:J-J-~ rtf • HJ H-= Hf if w ~ .'if '{e:-N
~"tv/(~ ~ H ~ ~ ~~J;;,
"""-J-' ~H ( 0() iA

~ a...

t1 it... s.J:: X of ~ ~ i fl ~G-. If ot:E: H~
.{~ (c<) .:a tt., ik..llo ~ X > ~~ if oc I H .fL., f;t~) ~ H ~
Hex -:+ H . fwrlhr ~ /H (O<p) :. ff' (p) ilt («¥.) to lt.-t o< ~ -?t (o<.- ')
I

o( (:

r

~

N)

~~~

.._

It ~ ~

' ./.H (o<)

I1A1;

1 N wiJJ.. /(~ ~ H, . 7l.w. -IH (N)
u-.. ..... 4
~IJ. fH ( 7) ~ .JL o<. E-N :.

fV

rJjf-1.

~ l: G.

If -c ~ ~ }k, ~rakur -f fH(G-) ~ ~(G-) ,.....J if -c ~r H
i.....b '14 ~t Hy-:. ~ -r ~....,t- ~ Hj
H*f for MI.. rE-G-.
~a, Hvt = H*7 t -fn- JL 1 f H ~ so H.,.=- H* H 4"~ H*- Ho(
cM1:t. ~ e N. ~ -c = -tH (o<) ca...J S· 3 k, }HnH.J .
LC.) Lek J> ~ .... t~ 4(,..J l..t G- ~w_ "'"~ '1- ~ p~~

h..*

~ ('M._, }' ) -= 1 .

f..--- s~

·f

S _;..

}W..;.k

1

~j s~

7L:v.:. ~ ;.,

& "

;-~

A.

4

Is- J _, ~tuz.... "

A.

G- af ~ ;"" ;.

S:JL.w ;-.s~

~~ JG:S} c.o ~ lr

~"J

= o

c.4UJ

.

t·

JL., ~ · S; ~ ;-s~

-1

~

G- w 1t.c..

'

7k. ~ ~ J ~ tt;. L. S:.~lo,.,

~ S·4-

(ii)

S:J &.w

1 G- if AAJ
If; ~ ,._~

f.A"':J; J~ .

\)

A.

IC31.- ·I1J£?".

~ G "-44 A/; ~t h~ s:J~
7k.. S:J!.u, t- s~~.c ~ {;. ·tUe. JL c~~.J;,.. ..;. G-.

V)

£wV'j

£4 r...~~-.c..J-~

;-s..Lsraut·

~ii) ~ Ev-e~ ;- .s~ ay G . u ~A.~ ~ .,._/; lr..ot DN- S"~ ;,-J~~
· \!"~ 7k """_.Lr of SJfo.w p-s~.s 1 C. ~
1 lk.,..J}. ·
.
-. .· 1'rrnr1- : <! l ~ & .LJ.:JA«-)1--. Lcl- X k
~~ ? G- ~
I XI -= "'A a-J ~ ~ r;.. 7Lt" I XsI -= p~ ":~ I• {, • ~ lk. ~v.L.r

=

Q...

s

7efK4ev.::tJ~

f

1-

G-

~

s

f

or

lr

J

,

X~ X
(I XI = J?')
.
~ 1tsa.
X . S~c:&. ( .... , ;) = t ~ ~ "~~ ;"'~-.,.. ~~
;"--..r «-re ~c:J,Ie, ~3 lk .s-.-.... l;.Jk,.t ;~r 1- f f'r .(l.a.cL v,._L_
1 -r -::. 0 _, 1_, .:l, . ;.. :) /'..._/ . So _ Jk ~,.u_. 1 f l.o }~ /,; f ~J
f /vw ~ &..at 0ML tra,wl,/-~ Cl-f~M.e.J; f, . WjJ_ ~ * I ~ "'t·
L~ x_, k c..~ "'"'-~ 1 1tc.. .~r'!!'~ frd~if-:..vJ;. clau ~.J ls.l
S 4 14 sf~c.l<.ur '1 X, ""- G-.. ~ /X,J -::.::. _ ca..J ./G: S } -= 'M ' ,
}>-n
~) , X, S =X 1 • So X1 io /t.-.. """~ 1 u..f~ i-~r.u.. ~ 'd S
/...,.c.c.. /S/ .,. {X,I = ;"'. 8.J: (....,, J>) = I -~~ /<T : Sj = ...., .
(f) ;

tt;:') ws

~!/J

~ ~c.u ~ ~~.J4-~

L.w

f/.t-o.. ; ""

~& Js 1 .· ~

fsJ =- ;"
\

I

."1

s

~

6..

sJw ;-.s~" 1 G-

r

l_ii ) ~) (iji) .
L.ei:: H k -.~:J .1~ 1 f!r • '" & f~ a/ S·2. {j'l J
~ ~ ~ J<At~ck~ h; H 1 lt... . '"erK~J.eJ;..J-~ ;H 1- G-. lJe. ''"',....,

?U<..J ILr, ~+..;.J~
t~~·J.wJ:J ~

cJL

equi;.

~

f 1
?- . f

1- ,H

~J,

c.~~

1- H;JK

~~ ~,

)U)

s~ I<

S<.c.AJ

-...

~ c.~

1 G-. 7L..
H UtA4'-sb 1J

s

Hl K .

w(;.4 cue.. ~11\..:uz.J ~

"" ~ ~

k4~

IE "

.,-H

7""k.. sJ- H I< ;,.
s-o -lt;:;,L ~ · ~ '1- H.) k ~ t;..

j,

Ld;- T ~~

d.to-H..cl:

1-ra~L&--n~..L

Q.

4 /t..

~~ ~

H > J<. ~ G- ~ i-.e. . lsJ: T eq.J-,;_ e~A,.clfJ ~ ~~.«~...t ~ ~ 1
·
~ ~iL ~.
~ ... G---=- , U 1-/"CK
~" 1""-. lin- H~K · ~

1

~j~ i- ;~~ .

't'E:T

1-k-u.

/G:H/-= "Cf:T /H-cl<: H/.. · .
:2..
.
,
1"k h.u-L.td /I-f -c K : H / ~ Jtc.. ~~ 1- Jk 1-ra#lJJ"J-("ve. ~fTYl~~
r;J ltz. r-cy~J-~ f ot K. N~ ."Cf ~Cl -c7 /i.e. ~ tt.. ..r~~ ~
"~ H ·f ~ ~ · -,
1
-·
1-' -c_, ~ H i ..e. t 7 E ~T· T~~ r 7 ~
.K we · ~cvkLe~~ s ·.» . L«: H ~a K L~ ~~~ 1 G- ~ {J; t E- Ga.J lsz.t T k
~· f ,-q,.J)\1"-Cr-4~ fE- ltc.. ~ c~ 1 H ~ I< ;_ G- • ~/ H I< = H 1 = IJc : Hs'"' 1< J
I G- : IN I -=- .2: I I< ~· H A /( I .

'ts

J

s

'1:

N6W · ~ H

=

S

~~

,

~~T

a..

S:Jl.,.w f-.!~ '1' G- a.....J kt I< l.c.

14tt.'J

p- s~ 1 G- . ~... I G- : S] i. /'~ 'k r c.-J .so ~ /,:J s-¥ -:1
/I< = '"n 1<. I ~ · p~ ~ f .,..,.. s~ -.:- ~ T. S~u. I< ~a a.. t-J"'""f"
i,C

~

11\.,..l;

/I< :

Jj~ p -s~ ~ G-.
::1 l&w

&,

s

p- s~

1

Q.~J

S '"C/"\ I<} -::. 1

G- ~

..So

· -a~~ <l~).
ItA..~

/I</

/< ~ S Y> ~ io

If

-= }S~/

~

1<.
.Io

i.o ~
/(

o.-

~: .

=- S T ~ ~"--JJj

~ G- . . ~ ~ (j_i/) •
~ ~

c-Jc.... H-=

= .S

B(t. 'r·S
Crr._ > .l'k. h."'--.l,.ec- ~ 1 S:~ln., . ;-s~J 1* G- ; w ~ tt.2.
l& . H/ , S~CA. ~ ~ JL ~~.h. fJ; s J,d (J.ii). If s ~ HT>'
. ftc."- /S: H-e" s I c:, ct.- ;~~~ f·~' ?- f. . If S ~ H~ :> lk....
({v)

Ntr(S)

I

o-l

/<.

.:... 4&

S"·S. ,

S, -::.

"C

S -c.- ' ~ H

u

~~

>

S

S:.~·lt.w p-.s~..r

eve

L!1 (.jii) • . ~c.L -r

S-= S1

a-.1 So

~J

~ . ,..o~:

~ {S : H T n

sI =

H ...... ~ H T S

'1- H ~ S

~)Co..elf:, ~ ~ c~

/ S ·. H -en SJ

E

. wi~ •

~~:J.L ~1' ; .

i .

/3~ S <t H

7 H.
= HS .

u.._ ;~ H s ~ ~ ~ci

F,r IL; ~C4j't~

7L.- 1 ::. 1..

,._._J

~ ~

r

""-CL ...

-r: ~ H

.J:Jlrw 's '"lk~""- ;...

"-"d

c~~ ;~J.

\p) A ~ ..;...~k..h. ~ .,_ J;w~ ~ ~ ~
-rc.~

~.
J •'r/ II p"'

44

I~

'

k~clu. }G-)~ ~ . G- kt14 J~.r

F" /,1- S· 2. \.yii ) , Q... S.:J ~ } - J~ S "' G- f.AA S~.s .
~ .L
P2.-> . •. > i'' =-Is J •
I

~

;

ltc.. crltu- ~J , /~ iLl-~ fn>"t

fL._

.D

.t~ tj
_. N~

FC/'r

sH

,._.,:t M- a.u.~
~~

1-/vJ: -.. . $~~

/G : HI -= ~
-= H J -= H f.-r

.t (: (j..

s 'f ,

~

-t

H
~

a.IL

a"-J ~~ ~.r

2::

~" ~

kn.tdLJ L
J

14 s~ X .. . 7I~ "n\.~ ~ /:;.k,..

sJ! =s

~ (A.~ "'A.:~ ~ '"'~

~~ ~

H

r~

~ a ·Lt2 E~~ 1\.~

Kr;. ts) -= ()

1.2.

44

I~ 2. ., •. , _:) )'L) .

"i4ot .:... H ) ~ tt:..~ 1:4~
wt1J.. G- - H .

,r-..

~Vof.f'j

1-

~~ oiJ ~ ("

~ ~ 1.~

')t.

~~c...h.

6r-

z:..t

( 74 .Sg-~·c..

'U>

~ ·~

f.,

_,

s~ "'J- ~ /, ..
~ lx) · .-..J 2. .,_(x) ~ JXI-=-

L

~

"'

H

Hj

;;,.

<t (;..

.>

~

'f

s~ ~

;,. ~.s · 'A~ · =·

G-

s

E

-t

G--H

(,;., c;. /,.,.k

~ ~ ~

M-J- (;, C. ~~ ~
~ ~ s~ ~ f-J~ ~ c;. . 1<&-(.s) ·;..
;-

s~

; - s~

1

G-. .

,

.

-J

. ~1tfPJJ. . . .
··~~~~

~

t..:t S·¥- (i ii) J M ~ s "r . , . So-a. -re-. 6-- .. ~..;.CA- M <I G ~ ~
-- 1~ ~ M ·.:.:.. M-c-'r ~ st . ~ J.L ~ (: r;. . ~ ~ -s; kG-(s) .
FN

I

C~o..:;~ !·tr3_

r

~

N

/I< : HJ

iA

I

s

lf

H 4~~ J<

I~ ~~~~-- ~ G 1 ft.,
~

Flrl"' . S

f4

S~ l.w ;-s~

,

J

/K: Nl · AK /k, "'....-L,-4 1 S:Jt.,..., ;-!~

A..J

,

rj '~ H a-A I< .J a.....J N=-NHts)-= .

b~

s: ~ (ji l· So IH : f\1 I ::. I k
/I< : HI ~ I ~ ; ":J ~, b .

~rif~ ~

s

Le~ s-- 6

74"' /' Ksj 1<

1 G- .r...J.. +LJ:. N ~ H ~I<
S:~fA.w p-s~ S "if' b-.) ~-

f·

NK (s) . ~" IH: N 1
tj H w I<. , H.a ~c::IWJ;r
L s:. tr ({V) • . . c,_L~ 4
:J

t:1trH!.

s~s

o...e

Ld 1<.. <~ &- ca..J k.t
u " Jjl&w; -s~ 1-

~.e

G/K

o... ·

: N J .... I ~ f

f:t~ f'-.$~

".J

S ~ 1< ~

o-

1-- G-.

SJ~

f -s~ tf k.

P~··

of f> _;
~J

13:;. ~· . s-·s.) · JI<S:K/=-/S!I<nS/

~d ;,~;~

cJfv:k ~ ~x ~ I<Sj I< ~ 6-/K ~ ~

sI

b~ 3·S"', .

J./.e-CJL, /G : I(SJ ~ ~~ -k />.
~ L,-- ~ ' /S . . . 1</ ;.. "- r~r 'II' .; ....~:J... II<:
1<1 "' ~
I s K : s I ~, s. s- ._~ IsK. : s I ..;. pn....... k ; .
. " ~"J : If I G- ' I< I .:. ;....:...... fr / , !A... S ~ I< • /f IG : 1<./
'

"'

~ ~ck., /6-:

/G; XSJ

(1...

r>

;~,... "~-

'; 2.•b .

~

1< s

s,.,

'::: G.

.

ld w- ~ a.."j s.d: ~ />~ , A TJ"-"-«~r iJ A-~
~ ;~ . -AJ~ J1, lL ;.... ~.
f.« -ro 1 k, lk ~i.--W'Jl...t~
s.d= 'J- f~ .
E~'1- p~W-e ~a- n. -= n-or "-rcr' , ~ n~
~

w . "'"f4T'

~ ~- ~..vkr

~

a.. ~!. n..~r . c.c.~

rl.di~

td- '1\,.
A~ S'Cit' -s~ ; ~ f"'Nt S t;. u. o."J ~~ s
trJ ~ /G-/ul.-e.. s"'-cl IU /.S/ . io a.. "t\r-J\.r.c.~r ;,._; J& :s /
~
A..
-c:r'- ~.,.,~r .
l.>f.e"" 't4r =;
~ A.. .r~k. ~
we. ~
h14~ /;s 14 ~ of "-- s.1~ ;-s~ wi..:.J.. ;.. ~~ 4"~~ llA
o.s1 - s~ . G £il C-Altll J . ,._ r.:r-I!O::<f ~ I G Iw =-/G I
or wloJ: io I~ S<A.-..llJ / G/-c:r' = 1. •
l.

R~:

tJtu.. 1-

ftc., I~

~ ~

"GJ'

..s~ ~ ~~

. o.lpLJui: . '. /..
'

~ ·-rr
I

u

I

c~~ w.1t (;.) _, ft.., ~~
a_

srad"~

~

-n:

oft.""

I.M).L~ L:J A.Of~tr-.t.~ -:.

Ctu..~

r.
~

t4

oJii)-.,...,.._~ .r.d"
L~"'"'-8.- S· t

~"

/(S/ 1<..

l.o

il- ~ ,._ot ~JA'-' b..J: }i

;~ _ s- b
1

oJ

f~

Afrlu.o ~

. Su

wdl..

sk.:J .

wi.IA ;

rcr~ '.:~

~ ~ J"t.~

4~

Sa- s~

-U:- s lu. ilA-:J ~- $~ oJ. (T_
'1 G/1<.. - ,_J . S n 1<. ~ t.tA Sr:o- ~~

IJ

G-

~ ~

:>

L-d I<

&

-<1

~J

~

&-

&/ - K .
~

aLo ~

le-~ s-. f?

.{

1\~. S~-J~ 1<

_
lk.. e.v<~ 7Ar~~~

·W ~ V-.J~ 6J- G- ~~.d.)~ K.

Fn- ~

H

~ "/ G- ~
iJ

a. ~-~

-- - -

-

~

~
~-

-- -- - -

1 r;.. S~aa.. I< ..q G- ~ KH ~ ·/K.H/-:::::. /kf. IH-. K" H/ . ~ }H = IC"- ffl

-r:r-s~

.-...

S"·~,

/1< / = /CYI-ct.

.

/~"'

lH:

Kr.H /_'- 1

-J H ~ I< .

'

\

·,

lA)

~ .s~

Let H k

1 G- .:.

if

-'!, •

H-= H0
(.ii)
\

~

tJe,

'f

~

f-r

·~

JM, "-~

H,

<J

HL

s

E

f

~ ~
~ iPt

·fL."'

;,t ~ -b; .r~

.s~

H.: s~c.k

~ Hr= G-.

<1 ·--

·cf

a...J ~

f

4"J ~

tl.a1; _

(..t)

if

H i.o

~~..1& 1:o Hl

G-.

Jv, ...~ .:.. G-

a..l,"-~ ~ G-

11

C-41L

11.-:J ~ •'f

p~ ~ lr

c.J... H

{«

(fv)

<l

H

CA.([

{H> tl}j
Iii)

~J

G-.

kk c:.JL H .r....J,..~ ~ G-

li)
.tc H.U

1

if .~J

l·

NCY (H)_ = H .

~ 'cj

ik i.o /,cY(t }~~

~ (} •

~~ · wilL

H s~ G-

L.

~J

~~

4-0

H />he. G- . ") . fl k"' G

>

~

1-1 ok. r;

~r~ -

1.

Le~ (,.f

Jm:r~~ ~ G-.

(J(,~~

H -<~ c;.. ~~ H sin.. G-- ~J aLo H t~~ G .) fv H =- Hs
. -::: {H) HI J -{tN- cJ.t f E- G- .
Sr...;r~ H .s~ · ~ ~v.E I:4C H ~ ~ot
~ ~ (;. lJe ~ ~ . /W:, (.1) _ ~ ~ lt:aJ: H (.. ncJ: Y\~
~ H2 • 71..¢~ H -4= H3 ~ s~ -s E- _H2.. _ Ik~ _ H1~ H/-= H, ~c.a- _
H,

<1

H2..

~~

H co.,.~tk k
~

'--

G

vJ,

~b.,.d{.

H
1

.

c~ ~~ _ to Ht .;.. 7.
~

H sin, t;,

74

:r-= ( H;) Hl}. ~ .

Jo

J.l

fnt

.G.

,;..,._,. !7

8~ H ~ H,

.t"t>

H <1 7 _ ~"~

Mr-~ H ;. n.~ ;tOY&~ ~
Hq

G-.

''

t;..

.

.s~ ~ G- J-:J.. ~ ·6~ s~ ~a Jio"~ .~ GSo we ~ ;c:.~,e, ltc.c, F e~ '1 s~

J..:~~~c.Jl:J

44

~

. ·.

:

---·----- ---

-

....

- -

I

....

'-·:-,
I

..